According to Gianluca, the Dirac function can only be approximated on a grid of finite size. This approximation is largely used for example in the immersed boundary method on structured grids, I strongly suggest to read the way C. Peskin proposed in a series of papers. You can get an example here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.453.2398&rep=rep1&type=pdf

I worked on a discretization of the Dirac on triangular grids, as explained in Sec.4.2.2 here https://www.researchgate.net/publication/229470711_2-D_Transmitral_flows_simulation_by_means_of_the_immersed_boundary_method_on_unstructured_grids

Article 2-D Transmitral flows simulation by means of the immersed bo...

Dear Alexander, usually in numerical scheme Dirac Delta is approximated using a 7-degree polynomial satisfying 6 conditions (two passages at two basis points with null-derivative, and a passage with a high value and null-derivative at central point); a source in a single cell could be a good idea where the polynomial can be defined; the value at central point can be computed using the 1-integral value.  Gianluca

According to Gianluca, the Dirac function can only be approximated on a grid of finite size. This approximation is largely used for example in the immersed boundary method on structured grids, I strongly suggest to read the way C. Peskin proposed in a series of papers. You can get an example here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.453.2398&rep=rep1&type=pdf

I worked on a discretization of the Dirac on triangular grids, as explained in Sec.4.2.2 here https://www.researchgate.net/publication/229470711_2-D_Transmitral_flows_simulation_by_means_of_the_immersed_boundary_method_on_unstructured_grids

Article 2-D Transmitral flows simulation by means of the immersed bo...

Gianluca, Filippo, thank you for prompt answers. I have finite size grid, and this polynomial approach seems to be natural for me. But I don't know, how wide I should distribute this polynom on a grid. Do you know any criteria for that?

I will read Peskin papers and will try to use his approach also. I hope, all approaches will give the same results.

Hi Alexander,

We did  discretization  for Dirac measure by Finite Difference where we had Dirac measure as source in our PDE. Please check the following paper, page 9, for dimension one and  page 11 for the two dimeansional case:

https://www.researchgate.net/publication/317387266_Multi_Phase_Q

Best

Data Multi Phase Q

Farid, thank you very much! I found a lot of useful references in you paper. Now I see that it is question of today's science, and there are a lot of options.

Dirac delta function's approximations are given in the book on page 61-62 by Van Der Pol and Bremmer (CUP, 1959). One of the useful function is the Dirichlet function that we have used in the book (page 124):

Instabilities of Flows and Transition to Turbulence-- Tapan K Sengupta (CRC Press, USA, 2012)

Hope this helps.

Depending on how the formulation of ur problem is. In so,e situation i use its Laplace or Fourier transform.

Yes, you can work in the transform plane and then the transform of delta function is like a white noise. But have thought of working in the transformed plane, Abdoulkadri? Who will pay for computing all the convolution integrals? Have you any estimate of error of such a procedure? That by itself will be a great research output. Instead working in physical plane with approximate delta function is much easier and comes with negligible error.

Dear doctor Sengupta,

Can you explain in a bit more detail, what do you mean with "working in physical plane"? Unfortunately, I do not have an access to Van Der Pol and Bremmer book (checked today). At the moment I plan to use any of the three functions, given in attached paper on a third page. In your answer did you mean such kind of approximations?

Never tried it before, but modeling the dirac-delta function by means of a normal distribution with a (very) small standard deviation could be an option?

If one starts from the weak formulation of the equations (that is, integrating the equations against a test function) then one can consider source terms like the Gaussian function with small variance mentioned in the previous answer and get a rigorous approximation of the Dirac function in the sense of distributions. Alternatively, one can also introduce a forcing term that is different to zero only at one node (this ends up to be identical to many of the previous proposals if the effective support of the Dirac delta approximation is narrow enough with respect to mesh resolution). The problem is that this approach will induce in most (not all) numerical methods spurious oscillations, so that in the end one just sees noise.

I am afraid working with Gaussian with narrow foot-print is not going to work for the following reason: Gaussian is a Hermitian function, which remains band-limited both in the physical and transformed plane. In contrast, the delta function in the physical plane excites all wave numbers in the transformed plane. Please use instead the Dirichlet function, given in my earlier response.

Dear researchers

I have the same problem.

I need to solve the 2-D Fokker Planck equation which is an initial condition parabolic differential equation with Delta Dirac initial condition,

I have no idea on how to simulate the Delta Dirac initial condition.

I am using finite difference method explicit in time.

In fact I have tried constructing the initial condition function(delta (x-x0)) as follows:

in the spatial mesh grid, I put the center of mass (equal to 1 in x0) and zero elsewhere on the mesh grid as shown in the picture.

But with this definition of the Delta Dirac function, the solution was incorrect (too many oscillations in the solution of PDE)

should I follow the same suggestion above to construct my initial condition?

Hello, Ali.

I recommend you this paper: Waldén J. On the approximation of singular source terms in differential equations. Numerical Methods for Partial Differential Equations: An International Journal. 1999. V. 16. P. 503–520.

In this paper author suggested different formulas, extending Diraq-delta function on 3, 5, 7 ... grid points. You can try some of them to see, how it works.

Note, not the absolute value of Diraq function should be equal 1, but integral over all space of Diraq function should be equal 1. @Ali_Namadchian

Dear Dr. Sebrekov

Thank you very much for the paper. it seems that in most papers the methods of the approximation of the Delta Dirac function are dependent on the numerical methods that are used to solve the PDE.

there is no guaranty that if a Delta Dirac approximation method works well with finite difference method, it also gives good result with finite element method, is it correct?

Ali Namadchian , no, I am sure that suggested in Waldén's paper approximations of Diraq delta function work well in finite difference method. I can not say something about finite elements method. Finite difference (implicit) is a simple golden standard in this area, I recommend to use it.